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By Gödel incompleteness, the fundamental axioms of mathematics actually don't quite pin down the world of mathematics the way we would perhaps like them to do.
by IngoBlechschmid 4y ago
By Gödel incompleteness, the fundamental axioms of mathematics actually don't quite pin down the world of mathematics the way we would perhaps like them to do. As a consequence (also a result by Gödel), there are actually many worlds of mathematics, each spelling out the basic notions of mathematics in slightly different ways.
I tried to survey this multiverse philosophy here: https://iblech.gitlab.io/bb/multiverse.html https://iblech.gitlab.io/bb/multiverse.html
- LudwigNagasena 4y agoWhat are “the fundamental axioms of mathematics”?
- IngoBlechschmid 4y agoI am sorry, I was being sloppy there. There are several systems of axioms we can use to base mathematics on. A common such is ZFC, Zermaelo–Fraenkel set theory with the axiom of choice. Among its axioms are assertions like "there is a set which is empty", "there is an infinite set" and "if A and B, then in particular A". Another is Martin-Löf type theory in one of its flavors, perhaps homotopy type theory.